Question

A rectangular box is to have a square base and a volume of 40 ft3. If the material for the base costs $0.36/ft2, the material for the sides costs $0.05/ft2, and the material for the top costs $0.14/ft2, determine the dimensions of the box that can be constructed at minimum cost.length ftwidth ftheight ft

Answer 1

Answer:

• length: 2 ft
• width: 2 ft
• height: 10 ft

Step-by-step explanation:

The total cost of the top and bottom is $0.36 + 0.14 = $0.50 per square foot.

The total cost of a pair of opposite sides is $0.05 +0.05 = $0.10 per square foot.

A minimum-cost box will have the cost of any pair of opposite sides be the same. Here, that means the box will have a side area that is 5 times the area of the top or bottom.

Since the base is square, that means the box is the shape of 5 cubes stacked one on the other. Each of those would be 8 ft³, so would have edge dimensions of ∛8 = 2 feet. The height is 5 times that, or 10 ft.

The box is 2 feet square by 10 feet high:

• length: 2 ft
• width: 2 ft
• height: 10 ft

_____

If you feel the need to write an equation, you can let x represent the edge length of the base. Then the cost of the top and bottom will be ...

  top&bottom cost = 0.50·x²

The height of the box is 40/x², so the cost of the four sides will be ...

  side cost = (0.05)(4x)(40/x²) = 8/x

This is minimized when the derivative of the sum of these costs is zero:

  cost = top&bottom cost + side cost

  cost = 0.50x² + 8/x

  d(cost)/dx = 1.00x -8/x² = 0

Multiplying by x², we get ...

  x³ -8 = 0

  x = ∛8 = 2 . . . . . . as above

  height = 40/x² = 40/4 = 10 . . . . . as above